A Montgomery–Hooley Theorem for sums of two cubes

Asympototic formulae are obtained for the variance associated with the distribution in arithmetic progressions of the numbers that are the sum of two positive cubes of integers.


Introduction
The sequence of numbers that are the sum of two positive cubes of integers, with or without multiplicities, has been studied from various perspectives. Hooley [5,7] showed that high multiplicities are rare, and his theme was taken up by Wooley [14] and Heath-Brown [4]. The distribution of the gaps between numbers that are sums of two cubes is the subject of recent work by Brüdern and Wooley [1]. Here we discuss the distribution in arithmetic progressions to provide an asymptotic formula for a certain variance that is analogous to the theorems of Montgomery [9] and Hooley [6] in the distribution of primes.
For N 1, Q 1 we are interested in establishing an asymptotic formula for (1.1) The average spacing between sums of two cubes suggests that such a formula should be available when Q is not too far from N 2/3 . In stating this theorem, and elsewhere in this memoir, we assert that whenever the letter ε occurs in a statement, then this statement is true for any positive real number assigned to ε. Constants implicit in the familiar symbols of Landau and Vinogradov depend on the particular choice of ε.
A result similar to Theorem 1.1 is available when sums of two cubes are counted without multiplicity. Thus, we define r 0 (n) = 1 whenever r (n) 1 and let r 0 (n) = 0 otherwise. As a consequence of the work of Hooley, we know that for a typical natural number n one has r (n) = 2r 0 (n). It is then natural to examine When studied on its own right, the ordering of sums of two cubes by size is certainly the most natural one. However, in the additive theory of numbers variables typically range over intervals independently, and it is then desirable to have at hand variance estimates that cover such situations. The following variant of Theorem 1.1 is directly applicable to the treatment of major arcs for additive problems involving cubes. Note that the range for Q where a valid asymptotic formula is obtained is larger than in Theorem 1.1. Let There is a rich literature on analogues of the Montgomery-Hooley formula for arithmetical functions. For example, the k-free numbers and other sifted sequences have been studied from this perspective (most recently Vaughan [13]) as well as other sequences that are reasonably well-behaved in arithmetic progressions (e.g. Hooley [8], Vaughan [12]). A common feature of existing work seems to be that the underlying sequence is fairly dense. In our situation, however, the expectation of r (n) grows like N 2/3 . We are not aware of other examples where a Montgomery-Hooley theorem is known and the growth rate for the expectation is as small as N θ with some θ < 1. Our method requires, at the very least, that the expectation grows faster than √ N . In fact, as we shall demonstrate in a sequel to this paper, it is possible to establish results of the type considered here for the sequence of numbers that are representable as the sum of a square and a positive k-th power. Yet, a Montgomery-Hooley theorem for sums of two biquadrates seems to require new ideas.
The core argument of this memoir is directed at an imperfect version of V where we withdraw from r (n) the Hardy-Littlewood expectation. To define the latter, let e(α) = exp(2π iα), write S(q, a) = q x=1 e(ax 3 /q) and, whenever T 1, put (1.4) Theorem 1.4 Suppose that 1 T N 3/5 and Q 1. Then In Sect. 3, we establish Theorem 1.4 by the Hardy-Littlewood method. This approach to Montgomery-Hooley formulae originates in the work of Goldston and Vaughan [3]. Our application prominently features the exponential sum that is less common than the cognate Weyl sum

Lemmata
We begin with some results concerning the function r (n).

Lemma 2.1
One has r (n) n ε and the number N(N ) of natural numbers n not exceeding N where r (n) = 2r 0 (n) satisfies Proof The first claim is a trivial consequence of a familar bound for the divisor function. It follows from Theorem 1 of Heath-Brown [4] that there are no more than O(N 4/9+ε ) natural numbers n N with r (n) > 2. Moreover, r (n) = 1 gives n = 2m 3 for some m ∈ N, and there are no more than O(N 1/3 ) such n. This confirms the second claim. The linear average of r (n) is an instance of the Lipschitz principle (Davenport [2]), and the formula for the quadratic average follows from the assertions of the lemma that we have already confirmed.
Our next lemma is the dual of the large sieve, and is a simple application of Theorem 1 of Montgomery and Vaughan [10].

Lemma 2.2
Suppose that M ∈ Z, N ∈ N and the η(t, c) are arbitrary complex numbers. Then Let κ be the multiplicative function defined for ν 0 and primes p by One routinely confirms the estimates Proof This follows from Lemmas 4.2, 4.3, 4.4 and Theorem 4.2 of Vaughan [11].
Then, by Lemma 2.8 of Vaughan [11], whenever |β| 1/2, we have We note that if j = 0 then this bound holds for all real numbers β. It is convenient to introduce the notation The next lemma is Theorem 4.1 of Vaughan [11].

Lemma 2.4
Suppose (t, c) = 1. Then, for j = 0 and 1 one has We are ready to investigate the exponential sum g(α), and begin by noting that by orthogonality and Lemma 2.1, one has 1 0 |g(α)| 2 dα N 2/3 . (2.5) Next, we turn to an analogue of Lemma 2.4 for the exponential sum g(α). Our result is relatively weak but suffices for our immediate needs. Improvements in the q-aspect would allow wider ranges for Q in Theorems 1.1 and 1.2. We define the sum and observe that the proof of Lemma 2.8 of Vaughan [11] provides the estimate where β denotes the distance of the real number β to the nearest integer. Further, we write Lemma 2.5 Let α ∈ R, q ∈ N and a ∈ Z. Then We write β = α − a/q and apply Lemma 2.4 to the sum over y to conclude that whence, on exchanging sum and integral, we infer We apply Lemma 2.4 again, this time to the sum over x. This yields and we find that Next, the substitution t = us in the inner integral produces is a special value of Euler's Beta function. In particular, we see that B 1 3 , 1 3 = 6C. Further, by Euler's summation formula, On collecting together, the lemma now follows.
It is convenient to estimate the mean square of the exponential sum h(α) = n N n −1/3 s(n; T ) e(αn).
Proof By orthogonality, Now let 1 M N . Then, by Lemmata 2.2 and 2.3, The lemma follows immediately, on summing over M = 2 ν N .
Proof We square out and recall the bound that is available from Lemma 2.6 and (2.9). For the cross product term, Cauchy's inequality together with Lemma 2.1 yields n N r (n) n −1/3 s(n; T ) and the lemma follows.

The proof of Theorem 1.4
We begin by squaring out (1.4). This yields

Now form the exponential sums
and .
For later use, we note that By (2.5) and Lemma 2.6, it follows that We now follow Section 3 of Goldston and Vaughan [3]. By Dirichlet's method of the hyperbola, Given a number r , on the right-hand side we separate terms with r | q from terms where r q. Then and H r (α) is the corresponding multiple sum with r q. On performing the inner summation in H r (α) we have For given b and r with (b, r ) = 1 we put Then αq bq/r − |β|q and so when q √ N , r q and |β| 1 2 and so We suppose that R and T satisfy and consider a typical interval M(r , b) associated with the element b/r of the Farey dissection of order R, namely, when 1 b r R and (b, r ) = 1, where r ± is defined by br ± ≡ ∓1 (mod r ) and R − r < r ± R and b ± is defined by b ± = (br ± ± 1)/r . We observe that We have We note also that F r (α) = 0 when r > N 1/2 . Hence, by (3.3), Hence Suppose that r √ N and define the major arc N(r , b) by Moreover, the same conclusion holds when α ∈ N(r , b) and r > N /R. Thus, by (3.3) and (3.6), where We now apply major arcs arguments to the function G(α). By (2.8) and (1.3), whence by (3.2) and (2.7), (3.10) We wish to use this within (3.9). Hence, suppose that α ∈ N(r , b) where 1 b r N /R and (b, r ) = 1. Then where in view of (3.10) we have (3.12) Here we estimate a typical summand when Hence, by Lemma 2.3, (2.6) and (2.7), we have the bounds We wish to sum this over all c and t with c/t = b/r . For a given t there is at most one c with 1 c t, (c, t) = 1 and If such a c exists we denote it by c 0 (t). The fractions c/t are spaced at least 1/t apart, modulo 1. Hence, by (3.13) and (3.14), Let T be the set of all t with 1 t T where c 0 (t) exists and satisfies c 0 (t We split into dyadic ranges. For 1 Z T let T(Z ) = T ∩ [Z , 2Z ). Then the fractions c 0 (t)/t are spaced (2Z ) −2 apart as t varies over T(Z ), and the c 0 (t)/t stay away from b/r at least (2r Z) −1 . By (3.13), we infer that We sum over Z = 2 ν T and combine the result with (3.15). This yields Proof We begin with the set of all α where D(α; r , b) (r 2/3 + T ) T ε holds. This set makes a contribution to the integral in question that in view of (3.7) does not exceed By This confirms the estimate proposed in Lemma 3.1.

Lemma 3.2 Suppose that (3.5) holds. Then
The reasoning leading to (3.19) applies here as well, and estimates the integral in question by

Lemma 3.3 Suppose that (3.5) holds. Then
Proof Within this proof, we abbreviate f (α, N 1/3 ) to f (α). The main observation is that in a suitable averaged sense one may replace |g(α)| by | f (α)| 2 . Let By orthogonality, Let J be the expression on the left-hand side of (3.20). Then, by (3.20), (3.21) We have where ψ h is the number of (x, y) ∈ Z 2 with x 3 − y 3 = h and 1 x, y N 1/3 . By (3.7), we have We now use (3.22) and sum over b to bring in Ramanujan's sum This produces Now note that ψ h 0. Hence, the classical bound |c r (h)| (r , h) suffices to conclude that the above expression does not exceed Observe that this bound is uniform in γ . We sum over r and deduce from (3.21) that The elementary bound K (γ ) min(N , γ −1 ) shows that the first factor on the right is O(log N ). This yields The lemma follows immediately.
The proof of Theorem 1.4 is now readily completed. We return to (3.11) and apply the elementary inequality |α + β| 2 2|α| 2 + 2|β| 2 to confirm that We multiply by |F r (α)|, integrate over N(r , b) and sum over 1 b r , (b, r ) = 1 and r N /R. The factor g − W is bounded by N 5/6+ε R −1/2 , as can be seen from Lemma 2.5. Then, by (3.9) and Lemmata 3.1, 3.2 and 3.3, we find that that is an instance of [11,Lemma 2.12]. This suggests to split the partial singular series s(n; T ) into S(t, c) 2 t 2 e(−cn/t) (4.4) and its complement S(t, c) 2 t 2 e(−cn/t). (4.5) and then see that A similar argument applies to a tail version of the sum in (4.4). This we define by This sum depends on n only modulo q. Therefore, with the convention concerning decorations of in mind, ‡ (N , Much as before, we find that and consequently, (4.9) By (4.8) and Cauchy's inequality, and then by orthogonality, Similarly, again via (4.8), we have From (4.9) we now see To bound Y 1 , write q = ts and reverse the order of summation. Via (2.2), we find that The estimation of Y 2 begins with Cauchy's inequality, applied to the sum over t. Then, by the same procedure that was successful with Y 1 , we find that This proves that We collect the results obtained so far in a single estimate. By where the very last bound has been confirmed earlier, in our estimate of Y 2 above. We now invoke the elementary evaluation together with (4.12) to confirm the bounds Equipped with the estimate obtained for the sum in (4.13), we deduce that This bound coupled with (4.11) implies Lemma 4.1.
The endgame begins by writing (1.1) as N , q, a) + B(N , q, a) Here the first summand in the error term is dominated by the last summand, so this reduces to the estimate claimed in Theorem 1.1.
Next suppose that N 17/20 < Q N , and take T = (N /Q) 4/9 . Then 1 T N 1/15 , Lemma 4.1 gives N 8/9+ε Q 4/9 and as above, Theorem 1.4 delivers Proceeding as before, we obtain Here, in the error term, the last summand dominates the others, so this also reduces to the estimate claimed in Theorem 1.1. In the range Q N 3/5 the theorem only asserts that V (N , Q) N 19/15+ε which follows from the bound for V (N , N 3/5 ) that we have already established.

The proof of Theorem 1.3
The proof of Theorem 1.3 is in principle simpler than that of Theorem 1.1 as it does not require the use of the convolution device embedded in the proof of Lemma 3.3. One follows the proceedings of Sect. 3 and replaces g(α) with f (α) 2 where f (α) = f (α; X ) is as in (1.6). Then, by Lemma 2.4, (2.4) and Lemma 2.3, the estimate given in Lemma 2.5 can be replaced by in the final stages of the proof of Theorem 1.4. For easier comparison, we put W * (α; q, a) = V 1 (α; q, a) 2 and then have, in the context of the last paragraph of Sect. 3, which can be used in place of the bound g − W N 5/6+ε R −1/2 and results in the possibility of taking a smaller value for R.
To be more precise, to effect the translation let N = 2X 3 , and take Then following the argument of Sect. 3 we have Then it follows that, cf. (3.8), Proceeding to use (5.1) in place of Lemma 2.5 we have The choice R = X 5/3 gives The analogue of Lemma 2.7 gives Thus Then, for 1 T Q X 3 , as in Lemma 4.1, one has * (X , Q, T ) Some justification is in order at this stage. Indeed, for a proof of (5.2), in the arguments of Sect. 4, one needs to replace the sum (4.1) by in which τ (n, X 3 ) substitutes the monotonic weight n −1/3 . While the function τ (n, X 3 ) is no longer initially decreasing, mundane asymptotic analysis reveals that is positive. Hence, there is a number n 0 such that τ (n, n) is decreasing for n n 0 . Furthermore, a rather more direct argument shows that τ (n, X 3 ) is also decreasing for X 3 < n N , and one immediately has τ (n, X 3 ) n −1/3 . These properties suffice to check that the initial phase of the proof of Lemma 4.1 still applies in the new set-up, and one confirms the bound as the appropriate counterpart of (4.11). The next step is to estimate and progress then depends on the bound n τ (n, . This is readily established by applying Abel summation on the interval b/q −1 n N , using the monotonicity of τ (n, X 3 ), while in the range n b/q −1 the trivial τ (n, X 3 ) n −1/3 is enough. Finally, we note that n τ (n, X 3 ) = 1 9 x X 3 This is weaker than the bound (4.14) but suffices to mimic the work after (4.11) successfully. However, the error term in the previous display now introduces an extra term of size X 2 into the final bound for * . This can be absorbed into the expression Q 4/3 T 2 + X 4 T −1 , and this proves (5.2). Now choose T = X 1/3 for Q X 9/4 , T = X 4/3 Q −4/9 for X 9/4 Q 2X 3 . Then, for Q X 9/4 , we find that U * (X , Q, T ) = 2Q X 2 + O X 11/3 + Q X 3/2 X ε and * (X , Q, T ) X 11/3+ε .
As in the proof of Theorem 1. Here the terms with n = m contribute We now temporarily suppose that N 3/5 Q N . Then E 0 Q N 4/9+ε , and Theorem 1.1 gives V (N , Q) Q N 2/3 . Hence E 1 Q N 5/9+ε . A short calculation now shows that the estimate in Theorem 1.2 follows from (6.1) and Theorem 1.1. As in the proof of Theorem 1.1, the case Q N 3/5 follows by considering the upper bound implied by the estimate for V 0 (N , N 3/5 ) that is already established.
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